Math Archive (14)
Which of the following graphs are equivalent?
1.
b
x
+
a
y
=
x
y
.
2.
y
=
x
−
a
b
x
.
3.
x
a
+
y
b
=
1
.
This problem is a submission of Problem Writing Party May 2016 .
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
For (1) also, same as (2) x = a
(1) and (2) are equivalent because but are not defined for x = a.
Log in to reply
@Rajen Kapur @No Kia Consider when b = 0 , (1) is two perpendicular straight line ( x = a or y = 0 ). (2) is only one straight line ( y = 0 ).
This question seems unreasonable because it cannot be properly answered without having some information about a and b. Equations 1 and 2 ARE equivalent unless a or b are zero; if that case really is to be considered, the question should make that clear, e.g. "Which of the following graphs are equivalent over all possible real values of a and b?" or something like that.
It's similar to being asked "Is A x + B y = C the equation of a line?" with only two possible answers, Yes and No, and having to choose No simply because if A and B are both zero, the equation represents either the empty set or the entire x y -plane. It's technically correct, but it feels like a misrepresentation.
Log in to reply
What do you mean? 2 graphs are equivalent only if every single point is the same. I mean every single point.
Log in to reply
Absolutely, but that is not what my comment was about. Let's use a specific example: suppose a = 2 and b = 3 . Then equation 1 is 3 x + 2 y = x y and equation 2 is y = x − 2 3 x . These equations ARE equivalent in every single point; in particular, there is no point with x = 2 on either graph. The same would be true for any other non-zero values of a and b .
Of course we always have to consider every permissible value of every variable in an equation; but coefficients/constants are a bit different. The only time equations 1 and 2 are not equivalent is when either a = 0 or b = 0 ; that was my point, that these two cases alone should not be sufficient to stop 1 and 2 from being called equivalent, unless we're specifically told to consider all possible values of a and b .
For ( 2 ) , x = a .
For ( 3 ) , x , y = 0 .
Although after some algebraic manipulation they look the same, the graph of ( 2 ) and ( 3 ) are missing some points.
They are all distinct graphs.